Issue

TROLL model currently compute leaf lifespan with Reich’s allometry (Reich et al. 1991). But we have shown that Reich’s allometry is underestimating leaf lifespan for low LMA species. Moreover simulations estimated unrealistically low aboveground biomass for low LMA species. We assumed Reich’s allometry underestimation of leaf lifespan for low LMA species being the source of unrealistically low aboveground biomass inside TROLL simulations. We decided to find a better allometry with Wright et al. (2004) GLOPNET dataset.

We tested different models starting from complet model Mcomp (with logged mean): \[ {LL_s}_j \sim \mathcal{logN}({\beta_0} + {\beta_1}_s*{LMA_s}_j^{{\beta_3}_s} - {\beta_2}_s*{Nmass_s}_j^{{\beta_4}_s},\,\sigma)\,\]

\[s=1,...,S_{=4}~, ~~j=1,...,n_s\] \[{\beta_i}_s \sim \mathcal{N}({\beta_i},\,\sigma_i)\,^I, ~~(\beta_i, \sigma, \sigma_i) \sim \mathcal{\Gamma}(0.001,\,0.001)\,^{2I+1}\] We tested models M1 to M9 detailed in following tabs to find the better trade-off between:

1. Complexity (and number of parameters)
2. Convergence
3. Likelihood
4. Prediction quality with Root Mean Square Error of Prediction (RMSEP)

RMSEP was computed by fitting a model without a dataset and evaluating it with the same dataset. RMSEP in results are mean RMSEP for each dataset. Results are shown for each models in eahc model tabs and summarized in Results tab.

Table 2: Models summary.

M	Model
M1	\(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - {\beta_2}_s*N,\sigma)\,\)
M2	\(LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\)
M3	\(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - {\beta_2}_s*N^{{\beta_4}_s},\sigma)\,\)
M4	\(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - N,\sigma)\,\)
M5	\(LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\)
M6	\(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\)
M7	\(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA,\sigma)\,\)
M8	\(LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s},\sigma)\,\)
M9	\(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s},\sigma)\,\)

LL graph

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - {\beta_2}_s*N,\sigma)\,\] Maximum likekihood of 11.1880728 and RMSEP of 12.346

Convergence

Model

\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of -0.5940783 and RMSEP of 11.402

Convergence

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - {\beta_2}_s*N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of 16.8245623 and RMSEP of 15.029

Convergence

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - N,\sigma)\,\] Maximum likekihood of 4.9012658 and RMSEP of 14.26

Convergence

Model

\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of -1.2010255 and RMSEP of 12.077

Convergence

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of 15.1501017 and RMSEP of 13.059

Convergence

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA,\sigma)\,\] Maximum likekihood of 4.1744354 and RMSEP of 36.089

Convergence

Model

\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s},\sigma)\,\] Maximum likekihood of -8.1187308 and RMSEP of 11.919

Convergence

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s},\sigma)\,\] Maximum likekihood of 9.4315134 and RMSEP of 17.553

Convergence

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Model M1 seemed to have shown the best trade-off between complexity (\(K=6\) parameters), convergence (see tab M1 ), likelihood, and prediction quality (see table 2). Figure 2 presents model prediction confidence interval and figure 3 compare Reich’s allometry and model M1 predictions with species functional traits used in TROLL. Nevertheless we will test other predictors gathered from TRY database following a model with the same response curve (see leaf lifespan ).

Table 2: Models likelihood and prediction quality.Root mean square errof of prediction (RMSEP) was computed by fitting a model without a dataset and evaluating it with the same dataset. RMSEP in results are mean RMSEP for each dataset.

	ML	RMSEP
M1	11.188	12.34647
M2	-0.594	11.40243
M3	16.825	15.02891
M4	4.901	14.26035
M5	-1.201	12.07721
M6	15.150	13.05924
M7	4.174	36.08934
M8	-8.119	11.91863
M9	9.432	17.55283

\[LL = 11.357*e^{0.008 *LMA -0.419*Nmass }\]

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